{
  "id": "2004.06671",
  "version": "v1",
  "published": "2020-04-14T17:18:21.000Z",
  "updated": "2020-04-14T17:18:21.000Z",
  "title": "On the Stability of Fourier Phase Retrieval",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $|\\widehat{f}|$. We study the stability properties of this problem in $L^p(\\mathbb{R}^n) \\cap L^2(\\mathbb{R}^n)$ for $1 \\leq p < 2$. The simplest result is as follows: if $f \\in L^1(\\mathbb{R}^n) \\cap L^2(\\mathbb{R}^n)$ has a real-valued Fourier transform supported on a set of measure $L < \\infty$, then for all all $g \\in L^1(\\mathbb{R}^n) \\cap L^2(\\mathbb{R}^n)$ $$ \\| f-g\\|_{L^2} \\leq 2\\cdot \\| |\\widehat{f}| - |\\widehat{g}| \\|_{L^2} + 30\\sqrt{ L} \\cdot \\|f-g\\|_{L^1}+ 2\\| \\Im \\widehat{g} \\|_{L^2}.$$ This is a form of stability of the phase retrieval problem for band-limited functions (up to the translation symmetry captured by the last term). The inequality follows from a general result for $f,g \\in L^p(\\mathbb{R}^n) \\cap L^2(\\mathbb{R}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-14T17:18:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier phase retrieval",
      "phase retrieval problem",
      "absolute value",
      "general result",
      "stability properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}